Do you remember when you were a child playing with picture blocks? A set of 12 or 20 cubes with part of six different pictures on each face. You could arrange them in a three by four or four by five rectangle to make each of the six pictures.

We saw a child playing with a set which had only four cubes which made six four by four pictures. She was throwing the blocks at random into their box, and suddenly gave a cry of delight - a picture had formed all by itself!

We immediately thought "What are the odds of that happening?", and of course we set about working it out.

So what are the odds of a picture appearing by chance? It does not matter which of the six pictures appears, and the picture can be any way up.

Place the first block at random in any corner of the box. There is a one in four chance that it is orientated correctly.

There are now three blocks to be placed. The first has a one in six chance that it shows the correct picture, a one in three chance that it is in the right corner, and a one in four chance that it is oriented correctly. (6 x 3 x 4 = 72)

The next has a one in six chance that it shows the correct picture, a one in two chance that it is in the right corner, and a one in four chance that it is oriented correctly. (6 x 2 x 4 = 48)

The last must be in the correct corner and has a one in six chance that it shows the correct picture and a one in four chance that it is oriented correctly. (6 x 4 =24)

Multiplying 4 x 72 x 48 x 24 gives 331,776, so the odds are one in 331,776 of a picture forming at random.